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324 XI ELEMENTARY SPECTRAL THEORY

(b) Each number A ^ 0 in the spectrum is an eigenvalue of u.

(c) For each A^Q in S, there is a unique decomposition of E into a
topological direct sum of two subspaces F(X), N(A) (also written F(l; u),
N(li uj) such that:

(i) F(/l) is closed, N(A) is finite dimensional;

(ii) «(F(A)) c F(l), awJ /Ac restriction of u - A • 1_E /o F(A) w a ///tear
homeomorphism of that space onto itself,

(iii) w(N(A)) c N(A) #/?£/ /Aere is a smallest integer k = k(X), called the
order ofl (also written k(l; «)), such that the restriction to N(/l) of(u — l- l_E)^fcisO.

(d) The eigenspace E(A) 0/ w corresponding to the eigenvalue X ^ 0 w
contained in N(A) (hence finite dimensional).

(e) IfX, fi are two different points 0/S, distinct from 0, /Ae/z N(/j) c F(/l).

(f) ^E w a Banach space, the function ( -* (u - C " IE)" *» wA/cA w defined
and analytic in C — S, Aoy a /?0/£ of order k(l) at each point A ^ 0 o/S.

Let /I ^ 0 be any complex number; as I'¹ u is compact, we can apply the
Riesz theory (Section 11.3). By (11.3.4), if A is not an eigenvalue of u, 1_E — A~^lu
is a linear homeomorphism of E onto itself, and the same is true of course
of M —A • 1_E= —A(1_E — A""¹^), i.e. A is regular for w, which proves b).
Suppose on the contrary A is an eigenvalue of u; then the existence of the
decomposition F(X) 4- N(A) of E with properties (i), (ii), (iii), follows from
(11.3.3), as well as (d) (E(X) is the kernel noted N_x in (11.3.3)). To end the
proof of (c), we need only show the uniqueness of F(/l) and N(/l). Suppose
there is a second decomposition E = F' + N' having the same properties, and
write v = u — 1-1_E. Then, any xeN' can be written x = y + z where
y e F(/l), z e N(A); by assumption there is h> 0 such that v^h(x) = 0, hence
v^h(y) = 0; as the restriction to F(A) of v^h is a homeomorphism by assumption,
7 = 0 and x e N(A). This proves that N' c N(A), and a similar argument
proves N(/l) c N'. Next, if .x = y + z e F' with y e F(/i), z e N(/l), we have
ifyc) = v\y), hence ^(F⁷) c F(A); but as i?(F) = F', this implies F' c F(X)₉

Denote by u_l9 u₂ the restrictions of u to F(A) and N(A), respectively. From
the relation (w₂ — ^' lN(A))^/t=^:0> it follows by linear algebra (A.6.10 and
A.6.12) that there is a basis of N(A) such that the matrix of u₂ — A • 1_N(A)with respect to that basis is triangular with diagonal 0; if d- dim (N(A)),
the determinant of u₂ — C * IN<A) is therefore equal to (A — C)^d and this proves
that «₂ ~ C ' IN<A> is invertible if C T^ A. Let us prove on the other hand that
^ui - C * IF(A) is invertible for C - A small enough: we can write Wj - f • 1_F(A) =
iY + (A — 0 • 1_F(A) with v₁ = z/j — A • 1_F(A). We know by (c) that t^ is inver-
tible; by (5.7.4), we therefore have \\v^^l(x)\\ ^ \\v^\\ • \\x\\ in F(A), which can
also be written K(;>c)|| > c - \\x\\ with c=||t?r¹||"¹. Now if C ^ 0 and



	324 XI ELEMENTARY SPECTRAL THEORY (b) Each number A ^ 0 in the spectrum is an eigenvalue of u. (c) For each A^Q in S, there is a unique decomposition of E into a topological direct sum of two subspaces F(X), N(A) (also written F(l; u), N(li uj) such that: (i) F(/l) is closed, N(A) is finite dimensional; (ii) «(F(A)) c F(l), awJ /Ac restriction of u - A • 1_E /o F(A) w a ///tear homeomorphism of that space onto itself, (iii) w(N(A)) c N(A) #/?£/ /Aere is a smallest integer k = k(X), called the order ofl (also written k(l; «)), such that the restriction to N(/l) of(u — l- l_E)^fcisO. (d) The eigenspace E(A) 0/ w corresponding to the eigenvalue X ^ 0 w contained in N(A) (hence finite dimensional). (e) IfX, fi are two different points 0/S, distinct from 0, /Ae/z N(/j) c F(/l). (f) ^E w a Banach space, the function ( -* (u - C " IE)" » wA/cA w defined and analytic in C — S, Aoy a /?0/£ of order k(l) at each point* A ^ 0 o/S. Let /I ^ 0 be any complex number; as I'¹ u is compact, we can apply the Riesz theory (Section 11.3). By (11.3.4), if A is not an eigenvalue of u, 1_E — A~^lu is a linear homeomorphism of E onto itself, and the same is true of course of M —A • 1_E= —A(1_E — A""¹^), i.e. A is regular for w, which proves b). Suppose on the contrary A is an eigenvalue of u; then the existence of the decomposition F(X) 4- N(A) of E with properties (i), (ii), (iii), follows from (11.3.3), as well as (d) (E(X) is the kernel noted N_x in (11.3.3)). To end the proof of (c), we need only show the uniqueness of F(/l) and N(/l). Suppose there is a second decomposition E = F' + N' having the same properties, and write v = u — 1-1_E. Then, any xeN' can be written x = y + z where y e F(/l), z e N(A); by assumption there is h> 0 such that v^h(x) = 0, hence v^h(y) = 0; as the restriction to F(A) of v^h is a homeomorphism by assumption, 7 = 0 and x e N(A). This proves that N' c N(A), and a similar argument proves N(/l) c N'. Next, if .x = y + z e F' with y e F(/i), z e N(/l), we have ifyc) = v\y), hence ^(F⁷) c F(A); but as i?(F) = F', this implies F' c F(X)₉ Denote by u_l9 u₂ the restrictions of u to F(A) and N(A), respectively. From the relation (w₂ — ^' lN(A))^/t=^:0> it follows by linear algebra (A.6.10 and A.6.12) that there is a basis of N(A) such that the matrix of u₂ — A • 1_N(A)with respect to that basis is triangular with diagonal 0; if d- dim (N(A)), the determinant of u₂ — C * IN<A) is therefore equal to (A — C)^d and this proves that «₂ ~ C ' IN<A> is invertible if C T^ A. Let us prove on the other hand that ^ui - C * IF(A) is invertible for C - A small enough: we can write Wj - f • 1_F(A) = iY + (A — 0 • 1_F(A) with v₁ = z/j — A • 1_F(A). We know by (c) that t^ is inver- tible; by (5.7.4), we therefore have \\v^^l(x)\\ ^ \\v^\\ • \\x\\ in F(A), which can also be written K(;>c)\|\| > c - \\x\\ with c=\|\|t?r¹\|\|"¹. Now if C ^ 0 and